Two Examples concerning Extendable and Almost Continuous Functions

The main purpose of this paper is to describe two examples. The first is that of an almost continuous, Baire class two, non-extendable function f : [0, 1] → [0, 1] with a Gδ graph. This answers a question of Gibson [15]. The second example is that of a connectivity function F : R → R with dense graph such that F−1(0) is contained in a countable union of straight lines. This easily implies the existence of an extendable function f : R → R with dense graph such that f−1(0) is countable. We also give a sufficient condition for a Darboux function f : [0, 1] → [0, 1] with a Gδ graph whose closure is bilaterally dense in itself to be quasi-continuous and extendable. 1 Definitions and Notation Our terminology is standard and follows [6]. We consider only real-valued functions of one or more real variables. No distinction is made between a function and its graph. A restriction of a function f : X → Y to a set A ⊂ X is denoted by f A. By R and Q we denote the set of all real and rational numbers, respectively, while I will stand for the interval [0, 1]. The closure of a set A ⊂ R is denoted by cl(A), its boundary by bd(A) and its diameter by